Similitude and Non–Dimensional Numbers

Similitude is a concept that relates the behavior of an object in a given flow field with its behavior in a different flow field under different operating conditions. This concept allows you to compare any two scenarios (simulations or experiments) and determine the similarities in the flow fields between them.

This concept becomes useful when there is a prototype of a vehicle, aircraft wing or any such application to be tested under laboratory conditions. It is not always feasible to test the full scale prototype owing to the large size or requirement of large scale flow conditions. Scaled models allow testing of a prototype prior to its production which can greatly increase the efficiency of the development process.

The following similarities need to be met in order to achieve similitude.

Geometric Similarity

Geometric similarity refers to similarity relations between the linear dimensions of the test scale model and the prototype. This means the model has the same shape as the prototype. More precisely, the model can be superimposed on the prototype by geometric operations like scaling, translation, rotation and reflection.

Kinematic Similarity

Kinematic similarity refers to the similarity of motion between the model and prototype in a fluid. This means that the flow streamlines are similar for the model and the prototype. More specifically, the velocities at corresponding points are in the same direction and are related in magnitude by a constant scale factor.

Dynamic Similarity

Dynamic similarity refers to the similarity between the force ratios at corresponding points and boundaries of the model and the prototype. This means that force distributions of the same type on the model and prototype have the same direction and are related in magnitude by a constant scale factor.

Figure 1. Similitude Between a Model and a Prototype

In order to analyse the above similarities certain non-dimensional numbers are used. The use of these numbers drastically simplifies the task of inferring the experimental data from the model and applying it to prototype design. Another advantage is that these numbers are independent of units of measurement and hence can be interpreted to suit any measurement system.

The most widely used non-dimensional numbers in fluid flow analysis are the following.

Reynolds Number

Reynolds number is the ratio of inertial (momentum) forces to viscous forces. It is an important parameter that can be used to determine the dynamic similarity between two cases of fluid flow or it can be used to characterize the flow regime in a fluid, this is, the laminar or turbulent nature of the flow. It is also used to study the transition between laminar and turbulent flows.

A laminar flow occurs at low Reynolds number when the viscous forces dominate resulting in a smooth velocity and pressure field where as a turbulent flow occurs at high Reynolds number where the inertial forces dominate the flow resulting in instabilities having a large range of time and length scales.

It is defined as (1)

Re = \frac{ρ u L}{μ}

where

$ρ$ is the density of the fluid
$u$ is the mean velocity of the flow
$L$ is the characteristic length
$μ$ is the dynamic viscosity of the fluid

Figure 2. Flow Characterization Based on Reynolds Number

The Reynolds number can also be expressed as the ratio of total momentum transfer to molecular momentum transfer.

The characteristic length is geometry dependent and defines the scale of physical system. For example, the characteristic length for a flow in a pipe with circular, square or annular cross section is taken as the hydraulic diameter of the pipe.

The hydraulic diameter for a pipe is defined as (2)

D_{H} = \frac{4 A}{P}

where

$A$ is the area of the cross section
$P$ is the wetted perimeter, that is, the perimeter of the pipe in contact with the flow

For a fluid moving between two plane parallel surfaces, where the width is much greater than the space between the plates, the characteristic dimension is twice the distance between the plates.

For an airfoil the characteristic length is considered to be the chord length.

There is no universal definition for a characteristic length. Any geometric length that significantly simplifies the governing equations when converted to their non-dimensional form is considered the characteristic length.

Mach Number

Mach number is the ratio of flow velocity to local speed of sound. It is a key parameter that characterizes the compressibility effects in a fluid flow.

When the Mach number is small (less than 0.3) the inertial forces experienced by the flow can be considered sufficiently small to not cause any changes to the fluid density. Under these conditions the compressibility effects of the fluid can be ignored.

It is written as (3)

M = \frac{u}{a}

The local speed of sound and hence the Mach number depend on the flow conditions, particularly on temperature and pressure of the fluid.

The Mach number is used to classify the flow regime into subsonic, transonic, supersonic and hypersonic flows, and they are defined as:

Subsonic flow $M < 0.8$
Transonic flow $0.8 < M < 1.0$
Supersonic flow $1.0 < M < 5.0$
Hypersonic flow $M > 5.0$

Knudsen Number

Knudsen number is the ratio of the molecular mean free path length to the representative physical length scale (characteristic length). It is a measure of rarefaction of the flow. Knudsen number determines whether continuum mechanics or statistical mechanics formulation should be used to model the flow.

It is expressed as (4)

K n = \frac{λ}{L}

where

$λ$ is the molecular mean free path understood as the mean distance travelled by a molecule between two successive collisions with other molecules
$L$ is the characteristic length

Prandtl Number

Prandtl number is the ratio of molecular diffusivity to thermal diffusivity in a fluid. When the Prandtl number is low it means that the heat diffuses quickly compared to momentum and vice versa.

It is written as (5)

P r = \frac{c_{p} μ}{k}

where

$c_{p}$ is the molar heat capacity of the fluid at constant pressure
$μ$ is the dynamic viscosity of the fluid
$k$ is the thermal conductivity of the fluid

Nusselt Number

The average Nusselt number is the ratio of convective heat transfer across a chosen boundary or surface to conductive heat transfer within a fluid. The convection heat transfer includes both advection and diffusion heat transfer.

It is defined as (6)

N u = \frac{h L}{k}

where

$h$ is the convective heat transfer coefficient
$L$ is the characteristic length
$k$ is the thermal conductivity of the fluid

A Nusselt number close to one is characteristic of a laminar flow whereas a large value corresponds to turbulent flow.

The local Nusslet number at a point at a distance x from the boundary is expressed as (7)

N u_{x} = \frac{h_{x} x}{k}

Pressure Coefficient

The pressure coefficient is the ratio of pressure difference to the dynamic pressure. It describes the relative pressure across a fluid field. It is expressed as (8)

C_{p} = \frac{p - p_{\infty}}{\frac{1}{2} ρ_{\infty} u_{\infty}^{2}}

where

$p$ is the point at which pressure coefficient is being calculated
$p_{\infty}$ , $ρ_{\infty}$ , $u_{\infty}$ are the free stream pressure, density and velocity, respectively

Lift and Drag Coefficients

The lift coefficient is the ratio of lift force to the dynamic force experienced by a body in a flow field.

It is expressed as (9)

C_{L} = \frac{L}{q A} = \frac{2 L}{ρ_{\infty} u_{\infty}^{2} A}

The drag coefficient is the ratio of drag force to the dynamic force experienced by a body in a flow field. It is expressed as (10)

C_{D} = \frac{D}{q A} = \frac{2 D}{ρ_{\infty} u_{\infty}^{2} A}

where

$L$ is the lift force experienced by the body
$D$ is the drag force experienced by the body
$q$ is the dynamic pressure of the fluid defined as $q = \frac{1}{2} ρ_{\infty} u_{\infty}^{2}$
$A$ is the reference area of the body

The reference area depends on the type of coefficient being measured. For automobiles and many bluff bodies the reference area while calculating the drag coefficient is taken as the projected frontal area. For airfoils the reference area is taken as the normal wing area.

The non-dimensional numbers used to determine similitude depend on the type of flow. For example, in a case of incompressible flow (without free surface), the Reynolds number can satisfy the similarity condition. If the flow is compressible, Reynolds number, Mach number and specific heat ratio are required in order to establish similarity.

It is not always possible to achieve absolute similitude between a model and prototype. It becomes even harder as you diverge from the prototype’s operating conditions, therefore it becomes important to focus on only the most important parameters.

The following table lists the most frequently encountered non-dimensional number, their description and applications.

Table 1. List of Frequenty Used Non-Dimensional Numbers
Name	Symbol	Numerator	Denominator	Formula	Applications
Reynolds number	$Re$	Intertial Force	Viscous force	$\frac{ρ u L}{μ}$	Fluid flow with viscous and inertial forces
Froude number	$F r$	Intertial Force	Gravitational force	$\frac{u}{\sqrt (g L)}$	Fluid flow with free surfaces
Weber number	$W e$	Intertial Force	Surface force	$\frac{ρ u^{2} L}{σ}$	Fluid flow with interfacial forces
Mach number	$M$	Local velocity	Local speed of sound	$\frac{u}{a}$	Gas flow at high velocity
Prandtl number	$P r$	Viscous diffusion rate	Thermal diffusion rate	$\frac{c_{p} μ}{k}$	Fluid flow with heat transfer
Nusselt number	$N u$	Convective heat transfer	Conductive heat transfer	$\frac{h L}{k}$	Fluid flow with heat transfer
Grashof number	$G r_{L}$	Buoyancy force	Viscous force	$\frac{g β (T_{s} - T_{\infty}) D^{3}}{μ^{2}}$	Fluid flow with natural convection
Rayleigh number	$R a_{x}$	(Buoyancy force) * (Viscous diffusion rate)	(Viscous force) * (Thermal diffusion rate)	$\frac{g β (T_{s} - T_{\infty}) x^{3}}{μ α}$	Buoyancy driven flow
Specific Heat ratio	$γ$	Enthalpy	Internal Energy	$\frac{c_{p}}{c_{v}}$	Compressible flow
Pressure Coefficient	$C_{p}$	Local pressure	Dynamic pressure	$\frac{p - p_{\infty}}{\frac{1}{2} ρ_{\infty} v_{\infty}^{2}}$	Pressure drop estimation
Lift Coefficient	$C_{L}$	Lift Force	Dynamic Force	$\frac{L}{q A}$	Aerodynamics, Hydrodynamics
Drag Coefficient	$C_{D}$	Drag Force	Dynamic Force	$\frac{D}{q A}$	Aerodynamics, Hydrodynamics
Skin friction Coefficient	$C_{f}$	Wall Shear Force	Dynamic Force	$\frac{τ_{w}}{q A}$	Aerodynamics, Hydrodynamics
Knudsen number	$K n$	Molecular mean free path	Characteristic length	$K n = \frac{λ}{L}$	Determination of applicability of continuum mechanics. i.e. suitability of AcuSolve for the application.